Effect of Individual Movement on Spatial Logistic Growth Dynamics
Trinh Thi Ngoc Mai, TAKASU Fugo
Department of Environmental Science,
Graduate School of Humanities and Sciences, Nara Women’s University
zat ten@cc.nara-wu.ac.jp

Abstract. Understanding population dynamics is fundamental in ecological modelling. Classical population models assume homogeneous mixing, ignoring spatial effects such as clustering and local interactions. Law et al. [2003] introduced a spatial logistic model by extending logistic growth to include local-scale processes such as dispersal-limited reproduction and short-range competition. Their model used individual-based simulations and moment equations to show how spatial interactions can cause substantial deviations from mean-field predictions.
In this study, we extend the spatial logistic framework of Law et al. by incorporating movement of individuals and examining its impact on both equilibrium population size and spatial structure. Each individual can undergo stochastic birth, death, and movement events governed by rates that depend on local density. Local interactions are defined by Gaussian kernels for competition and dispersal, while movement occurs at a fixed rate with a normally distributed dispersal range. We employ the Gillespie algorithm to simulate these stochastic processes and record both population sizes and spatial structures at each time step. To quantify the effects of movement, we conducted 100 simulation trials for each of five increasing movement rates, combined with varied dispersal ranges. Spatial structure was analyzed using the pair correlation function (PCF), which measures deviations from complete spatial randomness.
Our results show that movement greatly affects equilibrium population size and spatial pattern. At low movement rates, individuals remain near their birthplace, forming dense clusters that intensify local competition and increase the likelihood of local extinction. Increasing movement allows individuals to escape high-density regions, leading to reduced clustering, therefore higher survival rates and larger equilibrium populations. However, under long dispersal ranges, where offspring are already well spread, movement can have a neutral or even negative effect on local density by introducing anti-clusters. Notably, as movement increases for all dispersal ranges, the spatial pattern transitions toward complete spatial randomness. These findings demonstrate that movement plays two main roles: reducing crowding effects under limited dispersal and potentially homogenizing populations when dispersal is already broad. By incorporating movement, this study provides new insight into how mobility shapes population size and spatial patterning in ecological systems.
References
Richard Law, David J Murrell, and Ulf Dieckmann. Population growth in space and time: spatial logistic equations. Ecology, 84(1):252–262, 2003.